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Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form. (English) Zbl 1079.15005
The author introduces two canonical forms for Leonard pairs, the TD-D canonical form and the LB-UB canonical form. He gives several applications of his theory. As he proceeds through the paper he illustrates his results using two running examples which involve specific parameter arrays. Near the end of the paper he discusses how Leonard pairs correspond to the q-Racah polynomials and some related polynomials in the Askey scheme. At the end of the paper he presents some open problems concerning Leonard pairs.
The paper is divided into twenty eight sections: Introduction; Leonard systems; The relatives of a Leonard system; Leonard pairs and Leonard systems; Isomorphisms of Leonard pairs and Leonard systems; The adjacency relations; The eigenvalue sequences; The split sequences; A classification of Leonard systems; The notion of a parameter array; Parameter arrays and Leonard systems; The parameter arrays of a Leonard pair; The LB-UB canonical form, preliminaries; The LB-UB canonical form for Leonard systems; The LB-UB canonical form for Leonard pairs; How to recognize a Leonard pair in LB-UB canonical form; Leonard pairs A, A* with A lower bidiagonal and A* upper bidiagonal; Examples of Leonard pairs A, A* with A lower bidiagonal and A* upper bidiagonal; The TD-D canonical form, preliminaries; The TD-D canonical map; The TD-D canonical form for Leonard systems; The TD-D canonical form for Leonard pairs; How to recognize a Leonard pair in TD-D canonical form; Examples of Leonard pairs in TD-D canonical form; Leonard pairs A, A* with A tridiagonal and A* diagonal; How to compute the parameter arrays; Transition matrices and polynomials; Directions for further research.

MSC:
15A04 Linear transformations, semilinear transformations
15A21 Canonical forms, reductions, classification
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